Notes03 - Lecture 3 Univariate and Bivariate Probability...

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1 FRE 6083 – Quantitative Methods in Finance - Copyright F. Novomestky 2007 Lecture 3 – Univariate and Bivariate Probability Agenda É Simulation of Exponential Random Variable É Simulation of Uniform Random Variable É Simulation of Gumbel Random Variable É Non-Parametric Density Models É Bivariate Continuous Distributions É Multivariate Continuous Distributions É Bivariate Discrete Distributions É Multivariate Discrete Distributions 2 FRE 6083 – Quantitative Methods in Finance - Copyright F. Novomestky 2007 Simulation of Exponential Random Variable We use the inversion of CDF transformation from Lecture 2 to generate sample random variables or deviates that are independent and identically distributed (i.i.d.) according the following exponential distribution ( ) ( ) ( ) 1e x p X F xx H x λ ⎡⎤ =− ⎣⎦ We define the following relationship ( ) 1 X XFU = 3 FRE 6083 – Quantitative Methods in Finance - Copyright F. Novomestky 2007 Simulation of Exponential Random Variable ( ) () 1 x p exp 1 log 1 log 1 UX XU −= Excel workbook file Class03a.xls contains the sample simulation. R workspace contains the corresponding simulation using R. Let’s estimate the mean and variance of an exponential random variable.
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4 FRE 6083 – Quantitative Methods in Finance - Copyright F. Novomestky 2007 Simulation of Exponential Random Variable We’ll first derive the moment generating function for the exponential random variable and then compute the first two moments that are used to obtain the mean and variance. ( ) ( ) { } () 0 exp lim exp X b b MEX x dx ωω λλ ω λ ωλ →∞ = ⎡⎤ =− ⎣⎦ =< 5 FRE 6083 – Quantitative Methods in Finance - Copyright F. Novomestky 2007 Simulation of Exponential Random Variable 2 3 2 X X M M = ′′ = We set these derivatives equal to zero to obtain the first and second moments. 6 FRE 6083 – Quantitative Methods in Finance - Copyright F. Novomestky 2007 Simulation of Exponential Random Variable ( ) { } () {} 1 22 1 0 02 X X X ME X X σ == = We evaluate the simulations in terms of the theoretical versus estimated moments. 1 2 1 1 1 1 m i i m xi i xx m s m = = =
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7 FRE 6083 – Quantitative Methods in Finance - Copyright F. Novomestky 2007 Simulation of Uniform Random Variable We need the quantile function of a uniform random variable in order to simulate this random deviate ( ) 00 01 11 U u Fu u u u < = < ⎪⎩ ( ) U qp p = 8 FRE 6083 – Quantitative Methods in Finance - Copyright F. Novomestky 2007 Simulation of a Gumbel Random Variable We need the quantile function in order to simulate a Gumbel random variable ( ) ( ) log log X qp p ⎡⎤ =− ⎣⎦ 9 FRE 6083 – Quantitative Methods in Finance - Copyright F. Novomestky 2007 Non-Parametric Probability Models Histograms Naïve Estimator Kernel Estimator Empirical Distribution Function
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10 FRE 6083 – Quantitative Methods in Finance - Copyright F. Novomestky 2007 Histograms Oldest and most widely used density estimator.
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Notes03 - Lecture 3 Univariate and Bivariate Probability...

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